Minimum Area between a Semicircle and a Rectangle

The figure shows a semicircle of radius one and a horizontal line parallel to the diameter. What central angle corresponds to the minimum shaded area? This problem can be solved by either integration or trigonometry. The shaded area is equal to , where is the central angle. The minimum shaded area occurs when the central angle is .

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DETAILS

Moving the slider changes the height of the horizontal line and the area of the shaded region. The answer box shows the shaded area at any time. The plot displays the relationship between the shaded area and the central angle.
See [1], Chapter 8, Problem Plus 6.
Reference
[1] J. Stewart, Calculus: Early Transcendentals, 5th ed., Belmont, CA: Brooks/Cole, 2007.
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Related Curriculum Standards

US Common Core State Standards, Mathematics